Kind of looks like pascals triangle, i can't find the pattern the professor said

[mr_coffee]

Hello everyone. I have the following problem that can be solved using a traingle that looks similar to pascals triangle, but i must have have misunder stood the professors method of finding the next row. I have a final tomarrow and learning this technique would help but he told us on the last day of class so i didn't have time to ask him again what he did.The problem wants me to find out how many transitive, symetirc, and reflextive binary relations on S that has 5 elements. Well if somthing is transitive, symetric, and relfextive its just an equivlance relation, and a equivlance relation is just another way of saying, how many partitions exists in a set that has 5 elements? Well the answer is: 52.

Here is how the professor found it:
1
1 1
1 3 1
1 7 6 1
1 15 25 10 1

that's the 5th row down so 1 + 15 + 25 + 10 + 1 = 52, from what it looks like he is multiplying the first 1 in the 2nd row, by 2, then adding 1 = 3. then taking 3 * 2 + 1 = 7; but then i don't see how he is getting 6. I also see that he is taking 7*2 + 1 = 15, but I'm lost on how he found 25 and 10, any ideas? thanks.

 

[mr_coffee]

With the help of a friend i figured it out:
you just take the number * its positon add it to the left most number to get ur next number.
so the next row would be:
1 15*2 + 1 25*3+15 10*4 + 25 + 1*5 + 10
1 31 90 65 15 1